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Analysis of a viscous two-field gradient damagemodel

Part II: penalization limit

C. Meyer and L.M. Susu

Preprint 2016-13

ANALYSIS OF A VISCOUS TWO-FIELD GRADIENT DAMAGEMODEL

PART II: PENALIZATION LIMIT

C. MEYER§ AND L.M. SUSU§

Abstract. A viscous partial damage model is considered which features two damage variablescoupled through a penalty term in the stored energy functional. While the well-posedness of themodel was shown in a companion paper, this work analyses the behaviour of the model as thepenalization parameter tends to infinity. It turns out that in the limit both damage variables coincideand satisfy a classical viscous damage model.

Key words. Viscous damage evolution, energy balance, penalization

1. Introduction. This paper is concerned with a viscous gradient damage modelinvolving two damage variables which are connected through a penalty term in thestored energy functional. While the well-posedness of the model was investigated ina companion paper, cf. [21], this work addresses the limit analysis as the penalizationparameter tends to infinity. The damage model under consideration reads

(u(t), ϕ(t)) ∈ arg min(u,ϕ)∈V×H1(Ω)

E(t,u, ϕ, d(t)),

−∂dE(t,u(t), ϕ(t), d(t)) ∈ ∂Rδ(d(t)), d(0) = d0 a.e. in Ω

(P)

for almost all t ∈ (0, T ). Herein, d and ϕ denote the local and non-local damagevariable, respectively, and u stands for the displacement of the body occupying thedomain Ω ⊂ RN , N = 2, 3. Moreover, the stored energy E : [0, T ] × V × H1(Ω) ×L2(Ω)→ R is given by

E(t,u, ϕ, d) :=1

2

∫Ω

g(ϕ)Cε(u) : ε(u) dx− 〈`(t),u〉V +α

2‖∇ϕ‖22 +

β

2‖ϕ− d‖22,

(1.1)where V := W 1,2

D (Ω), ε = 1/2(∇+∇>) is the linearized strain, C the elasticity tensor,and ` the load applied to the body. The function g describes the influence of damageon the elastic behavior of the body. Furthermore, α > 0 is a weighting parameterfor the gradient regularization and β > 0 stands for the penalization parameter. Theviscous dissipation functional Rδ : L2(Ω)→ [0,∞] appearing in (P) is defined as

Rδ(η) :=

r∫

Ωη dx+ δ

2‖η‖22, if η ≥ 0 a.e. in Ω,

∞, otherwise,(1.2)

where r > 0 stands for the threshold value which triggers the damage evolution andδ > 0 is the viscosity parameter. For a more detailed description of the model as wellas its motivation, we refer to [21, Section 2] and [5,6]. Throughout the paper, we willrefer to (P) as “two-field model” or “penalized damage model”.

Damage models containing two different damage variables coupled through a penal-ization of the energy functional are frequently used for numerical simulations, cf.

§Faculty of Mathematics, Technische Universität Dortmund, Dortmund-Germany.

1

e.g. [5, 6, 20, 25, 26, 28, 31]. With this work and the companion paper [21], we showthat this approach is mathematically justified for two reasons. First of all, the two-field model (P) admits a unique solution in terms of the two damage variables andthe displacement field, and all three quantities depend Lipschitz continuously on thedata, i.e., the applied loads, so that the problem is well-posed. These issues are ad-dressed in the companion paper [21]. Secondly, as we will show in this paper, inthe limit β → ∞, both damage variables become equal and the resulting single-fielddamage model falls into the category of classical viscous, partial damage models, asfor instance introduced in [10].Let us put our work into perspective. Numerous damage models have been addressedby many authors under different aspects, among these viscous, i.e., rate-dependentmodels, see e.g. [1–3, 9], but also rate-independent damage models, cf. for instance[7,15,17–19,24]. Especially in the discussion of the latter, the so-called energy balanceplays a crucial role and various notions of solutions are based on it, such as forexample global energetic and semi-stable solutions, see [23] for an overview. Theenergy balance (sometimes appearing in a weaker form as an energy inequality) isalso an essential tool for the limit analysis for vanishing viscosity as performed in[17]. This passage to the limit leads to another notion of solutions, the so-calledparametrized solutions, see [17, Section 5] and [22]. The energy balance also turnsout to be very useful for our limit analysis, when β → ∞. To be more precise, wereformulate (P) as an equivalent energy identity. This allows for a passage to thelimit via lower semicontinuity arguments. The resulting inequality leads in turn toan energy-dissipation balance for the limit problem, by employing a well-known chainrule argument. This energy balance ultimately yields the single-field damage modelmentioned above.The paper is organized as follows. Section 2 collects the notations and standingassumptions, as well as known results from the companion paper [21], which areneeded in the present paper. In Section 3 we derive the energy inequality mentionedabove. Section 4 is devoted to the passage to the limit β →∞ in the energy inequality.In Section 5, we will reformulate the latter as a PDE system involving only one singledamage variable. By means of a time-discretization we then show in Section 6 thatthe damage variables associated with the penalized model (P) are essentially boundedin time and space by a constant independent of the penalty parameter β so that thisbound carries over to the damage variable in the limit. Based on this result, wetransform the single-damage model from Section 5 into an equivalent PDE systemthat coincides with the viscous model in [17] and constitutes a classical, viscous partialdamage model.

2. Notation, Standing Assumptions, and Known Results. Throughoutthe paper, C denotes a generic positive constant. If X and Y are two linear normedspaces, the space of linear and bounded operators from X to Y is denoted by L(X,Y ).The dual of a linear normed space X will be denoted by X∗. For the dual pairingbetween X and X∗ we write 〈., .〉X and, if it is clear from the context, which dualpairing is meant, we just write 〈., .〉. By ‖·‖p we denote the Lp(Ω)−norm for p ∈ [1,∞]and by (·, ·)2 the L2(Ω)−scalar product. If X is compactly embedded in Y , we writeX →→ Y . In the rest of the paper N ∈ 2, 3 denotes the spatial dimension. Bybold-face case letters we denote vector valued variables and vector valued spaces.Definition 2.1. For p ∈ [1,∞] we define the following subspace of W 1,p(Ω):

W 1,pD (Ω) := v ∈W 1,p(Ω) : v|ΓD = 0,

2

where ΓD is a part of the boundary of the domain Ω, see Assumption 2.2 below. Thedual space of W 1,p′

D (Ω) is denoted by W−1,pD (Ω), where p′ is the conjugate exponent

of p. If p = 2, we abbreviate V := W 1,2D (Ω).

Before we turn to our assumptions on the data, we summarize the symbols often usedthroughout the paper in Table 2.1 for convenience of the reader.

Table 2.1Functionals, operators and variables

Symbol Meaning DefinitionI Reduced energy functional Definition 3.1Rδ Viscous dissipation functional (1.2)u Displacementϕ Nonlocal damaged Local damageE Energy functional without penalty Definition 4.10I Reduced energy functional without penalty (4.16)R1 Dissipation functional in the limit (5.10)Rδ Viscous dissipation functional in the limit Definition 4.13

Let us now state our standing assumptions. We emphasize that these are tacitlyassumed in all what follows without mentioning them every time. We begin with thesmoothness of the computational domain.

Assumption 2.2. The domain Ω ⊂ RN , N ∈ 2, 3, is bounded with Lipschitzboundary Γ. The boundary consists of two disjoint measurable parts ΓN and ΓD suchthat Γ = ΓN ∪ ΓD. While ΓN is a relatively open subset, ΓD is a relatively closedsubset of Γ with positive measure.

In addition, the set Ω ∪ ΓN is regular in the sense of Gröger, cf. [11]. That is, forevery point x ∈ Γ, there exists an open neighborhood Ux ⊂ RN of x and a bi-Lipschitzmap (a Lipschitz continuous and bijective map with Lipschitz continuous inverse)Ψx : Ux → RN such that Ψx(x) = 0 ∈ RN and Ψx

(Ux ∩ (Ω ∪ ΓN )

)equals one of the

following sets:

E1 :=y ∈ RN : |y| < 1, yN < 0

,

E2 :=y ∈ RN : |y| < 1, yN ≤ 0

,

E3 := y ∈ E2 : yN < 0 or y1 > 0 .

A detailed characterization of Gröger-regular sets in two and three spatial dimensionsis given in [12].

Assumption 2.3. The function g : R→ [ε, 1] satisfies g ∈ C2(R) and g′, g′′ ∈ L∞(R)with some ε > 0. With a little abuse of notation the Nemystkii-operators associatedwith g and g′, considered with different domains and ranges, will be denoted by thesame symbol.

Assumption 2.4. The fourth-order tensor C ∈ L∞(Ω;L(RN×Nsym )) is symmetric anduniformly coercive, i.e., there is a constant γC > 0 such that

C(x)σ : σ ≥ γC|σ|2 ∀σ ∈ RN×Nsym and f.a.a. x ∈ Ω, (2.1)3

where | · | denotes the Frobenius norm on RN×N and the symbol “ : ”stands for thescalar product inducing this norm.

Assumption 2.5. For the applied volume and boundary load we require

` ∈ C1([0, T ];W−1,pD (Ω)),

where p > N is specified below, see Assumption 2.7.1 and Assumption 5.4.

Moreover, the initial damage is supposed to satisfy d0 ∈ L2(Ω).

Our last assumption concerns the balance of momentum associated with the energyfunctional in (1.1). For its precise statement we need the following

Definition 2.6. For given ϕ ∈ L1(Ω) we define the linear form Aϕ : V → V ∗ as

〈Aϕu, v〉V :=

∫Ω

g(ϕ)Cε(u) : ε(v) dx.

The operator Aϕ, considered with different domains and ranges, will be denoted by thesame symbol for the sake of convenience.

Assumption 2.7. For the rest of the paper we require the following:

1. There exists p > N such that, for all p ∈ [2, p] and all ϕ ∈ L1(Ω), the operatorAϕ : W 1,p

D (Ω)→W−1,pD (Ω) is continuously invertible. Moreover, there exists

a constant c > 0, independent of ϕ and p, such that

‖A−1ϕ ‖L(W−1,p

D (Ω),W 1,pD (Ω)) ≤ c

holds for all p ∈ [2, p] and all ϕ ∈ L1(Ω).2. The penalization parameter β is sufficiently large, depending only on the given

data, see [21, Eq. (3.33)].

Remark 2.8. (i) The critical assumption is Assumption 2.7.1. If N = 2, thenthis condition is automatically fulfilled, see [21, Lemma 3.3]. The situation changeshowever, if one turns to N = 3. In this case this assumption can be guaranteed byimposing additional and rather restrictive conditions on the data, in particular on theellipticity and boundedness constants associated with C and g, see [21, Remark 3.11]for more details. However, as explained in [21, Remark 3.12], one could alternativelymodify the energy functional in (1.1) by replacing ‖∇ϕ‖22 with its H3/2-seminorm.This would allow us to drop Assumption 2.7.1 in the three dimensional case, too.However, we have chosen not to work with the H3/2-seminorm, as the associatedbilinear form is difficult to realize in numerical computations.

(ii) For convenience of the reader, we mention here that in [21, Eq. (3.33)] one requiresβ > α+ C Ck, where the constant C > 0 depends on the supremum norm of ` and ˙,the Lipschitz constants of g and g′, ‖C‖L∞(Ω;L(RN×Nsym )) and embedding constants, whilethe fixed value of Ck > 0 depends on k and p, see the proof of [21, Lemma 3.18].

As an immediate consequence of Assumption 2.7.1 and the regularity of ` in Assump-tion 2.5 one can introduce the following

Definition 2.9. We define the operator U : [0, T ]×H1(Ω)→W 1,pD (Ω) by U(t, ϕ) :=

A−1ϕ `(t). We will frequently consider U with different ranges and domains, but denote

it by the same symbol.4

Thanks to Assumptions 2.5 and 2.7.1 there exists a constant c > 0, independent of tand ϕ, such that

‖U(t, ϕ)‖W 1,pD (Ω) ≤ c ∀ (t, ϕ) ∈ [0, T ]×H1(Ω), (2.2)

which will be frequently used in the sequel.

In the rest of this section we recall some known results and definitions from thecompanion paper [21] that will be used in the upcoming analysis.

Lemma 2.10 (Global Lipschitz continuity of U , [21, Proposition 3.8]). Let p > 2and r ∈

[2p/(p− 2),∞

]be given. Then there exists L > 0 such that for all ϕ1, ϕ2 ∈

H1(Ω) ∩ Lr(Ω) and all t1, t2 ∈ [0, T ] it holds

‖U(t1, ϕ1)− U(t2, ϕ2)‖W 1,πD (Ω) ≤ L(|t1 − t2|+ ‖ϕ1 − ϕ2‖r), (2.3)

where 1/π = 1/p+ 1/r.

Lemma 2.11 (Fréchet differentiability of U , [21, Proposition 5.6]). It holds U ∈C1([0, T ]×H1(Ω);V ) and at all t ∈ [0, T ] and ϕ, δϕ ∈ H1(Ω) we have

∂tU(t, ϕ) = A−1ϕ

˙(t) ∈W 1,pD (Ω), (2.4a)

Aϕ(∂ϕU(t, ϕ)(δϕ)

)= div

(g′(ϕ)(δϕ)Cε(U(t, ϕ)

)in V ∗, (2.4b)

where div : L2(Ω;Rn×nsym )→ V ∗ denotes the distributional divergence. Moreover, thereexists a constant c > 0, independent of t and ϕ, such that

‖∂tU(t, ϕ)‖W 1,pD (Ω) ≤ c ∀ (t, ϕ) ∈ [0, T ]×H1(Ω). (2.5)

In order to state the Euler-Lagrange equations associated with the energy mini-mization in (P), let us further define the mappings B : H1(Ω) → H1(Ω)∗ andF : [0, T ]×H1(Ω)→ H1(Ω)∗ by

〈Bϕ,ψ〉H1(Ω) :=

∫Ω

α∇ϕ · ∇ψ + βϕψ dx, ϕ, ψ ∈ H1(Ω), (2.6)

〈F (t, ϕ), ψ〉H1(Ω) :=1

2

∫Ω

g′(ϕ)Cε(U(t, ϕ)) : ε(U(t, ϕ))ψ dx, ϕ, ψ ∈ H1(Ω). (2.7)

Note that F is well defined because of the Sobolev embedding H1(Ω) → Ls(Ω) withs = 6 for N = 3 and s <∞ for N = 2, in combination with Assumption 2.7.1.

Lemma 2.12. The mapping F possesses the following properties:

• [21, Lemma 3.17] It is Lipschitzian in the following sense: Let r ≥ 2p/(p−2)and 1/s + 2/p + 1/r = 1. Then, for all t1, t2 ∈ [0, T ], all ϕ1, ϕ2 ∈ H1(Ω) ∩Lr(Ω) and all ψ ∈ Ls(Ω), there holds

|〈F (t1, ϕ1)− F (t2, ϕ2), ψ〉H1(Ω)| ≤ C(‖ϕ1 − ϕ2‖r + |t1 − t2|

)‖ψ‖s, (2.8)

with a constant C > 0 independent of (ti, ϕi)i=1,2.5

• [21, Lemma 5.9] It is continuously Fréchet differentiable from [0, T ]×H1(Ω)to H1(Ω)∗, and for all (t, ϕ) ∈ [0, T ] ×H1(Ω) and all (δt, δϕ) ∈ R ×H1(Ω)we have

〈F ′(t, ϕ)(δt, δϕ), z〉H1(Ω) =1

2

∫Ω

g′′(ϕ)(δϕ)Cε(U(t, ϕ)) : ε(U(t, ϕ))z dx

+

∫Ω

g′(ϕ)Cε(U(t, ϕ)) : ε(U ′(t, ϕ)(δt, δϕ)

)z dx

(2.9)for all z ∈ H1(Ω).

• [21, Lemmas 3.17, 3.18] For its partial derivative w.r.t. ϕ there holds

|〈∂ϕF (t, ϕ)z, z〉H1(Ω)| ≤ k‖z‖22 + c(k)‖z‖2H1(Ω) (2.10)

for all z ∈ H1(Ω), all k > 0, and all (t, ϕ) ∈ [0, T ]×H1(Ω), where c : R+ →R+ is a monotonically decreasing function, which tends to 0 as k →∞.

With the mappings B and F at hand we can characterize the solution to the energyminimization in (P) as follows:

Lemma 2.13 (Energy minimizer, [21, Prop. 3.14, Thm. 3.20]). For every (t, d) ∈[0, T ]× L2(Ω), the optimization problem

min(u,ϕ)∈V×H1(Ω)

E(t,u, ϕ, d)

admits a unique minimizer (u, ϕ) ∈ W 1,pD (Ω) ×H1(Ω) characterized by u = U(t, ϕ)

and ϕ = Φ(t, d), where Φ : [0, T ] × L2(Ω) → H1(Ω) is defined by Φ(t, d) := (B +F (t, ·))−1(βd).

Lemma 2.14 (Global Lipschitz continuity of Φ, [21, Eq. (3.35)]). There exists aconstant K > 0 such that

‖Φ(t1, d1)−Φ(t2, d2)‖H1(Ω) ≤ K(|t1−t2|+‖d1−d2‖2

)∀ t1, t2 ∈ [0, T ], d1, d2 ∈ L2(Ω).

Lemma 2.15 (Fréchet differentiability of Φ, [21, Prop. 5.12]). The solution operatorΦ is continuously Fréchet differentiable from (0, T )×L2(Ω) to H1(Ω). Moreover, forall (t, d) ∈ [0, T ]×L2(Ω) and all (δt, δd) ∈ R×L2(Ω) its derivative solves the followinglinearized equation

BΦ′(t, d)(δt, δd) + F ′(t, ϕ)(δt,Φ′(t, d)(δt, δd)

)= βδd, (2.11)

where we use the abbreviation ϕ := Φ(t, d).

Finally we turn our attention to the differential inclusion in (P). First note that thefunctional E is partially Fréchet differentiable w.r.t. d on [0, T ]×V ×H1(Ω)×L2(Ω),and its partial derivative is given by

∂dE(t,u, ϕ, d) = β(d− ϕ). (2.12)

Therefore, in view Lemma 2.13, (P) reduces to the following evolutionary equation

− β(d(t)− Φ(t, d(t))

)∈ ∂Rδ(d(t)) f.a.a. t ∈ (0, T ), d(0) = d0. (2.13)

6

As shown in [21, Lemma 3.22], this equation is equivalent to the following non-smoothoperator differential equation:

d(t) =1

δmax−β

(d(t)− Φ(t, d(t))

)− r, 0 f.a.a. t ∈ (0, T ), d(0) = d0. (2.14)

This handy reformulation of (P) and (2.13), respectively, is a main advantage ofthe penalty-type regularization of partial damage models and could provide a usefulstarting point for a numerical solution of (P). We end this section by recalling themain result of the companion paper:

Theorem 2.16 (Existence and uniqueness for the penalized damage model, [21,Thm. 5.13]). There exists a unique solution (u, ϕ, d) of the problem (P), satisfyingu ∈ C1([0, T ];V ), ϕ ∈ C1([0, T ];H1(Ω)), d ∈ C1,1([0, T ];L2(Ω)), which is uniquelycharacterized by the following system of differential equations:

−div g(ϕ(t))Cε(u(t)) = `(t) in W−1,pD (Ω) (2.15a)

−α∆ϕ(t) + β ϕ(t) +1

2g′(ϕ(t))C ε(u(t)) : ε(u(t)) = βd(t) in H1(Ω)∗ (2.15b)

d(t)− 1

δmax−β(d(t)− ϕ(t))− r, 0 = 0, d(0) = d0. (2.15c)

for every t ∈ [0, T ].

Note that, thanks to the definition of B and F , the equations in (2.15a) and (2.15b)are just equivalent to u(t) = U(t, ϕ(t)) and ϕ(t) = Φ(t, d(t)), respectively.

3. Energy Balance. As seen in Theorem 2.16, for a given β > 0 sufficientlylarge, there exists a unique solution to (2.14), which we denote by dβ to indicate itsdependency on the parameter β. The other components are uniquely determined bydβ through ϕβ = Φ(·, dβ(·)) and uβ = U(·, ϕβ(·)). The purpose of this section is toderive a characterization of the local damage dβ , which allows us to find an estimateof the form ‖dβ‖X ≤ C for all β > 0, where X is a suitable reflexive Banach spaceand C > 0 is a constant independent of the penalty parameter β. Such an estimatewill then enable us to pass to the limit in the penalized damage model as β → ∞,see Section 4 below. As seen in (2.13) and (2.14) above, there are various ways todescribe the evolution of the local damage. However, all these descriptions have thedisadvantage of containing the term β(dβ − ϕβ), which is not necessarily uniformlybounded w.r.t. β in suitable spaces that allow for a passage to the limit. Our aim istherefore to find an alternative description of the evolution of the local damage, whichonly contains expressions that are bounded w.r.t. β. Such a description is given bythe energy-dissipation balance in Proposition 3.5 below.

For the rest of this section we drop the index β to shorten the notation. As alreadyindicated above, the displacement u and the nonlocal damage ϕ are uniquely deter-mined by the local damage variable d so that it is reasonable to reduce the wholesystem to the variable d only. For this purpose we define the following:

Definition 3.1. The reduced energy functional I : [0, T ]× L2(Ω)→ R is given by

I(t, d) := E(t,U(t,Φ(t, d)),Φ(t, d), d).

7

The reduced energy functional will be a key ingredient for deriving the energy balance.On account of (1.1) and Definitions 2.6 and 2.9 it can be rewritten as

I(t, d) =1

2〈AΦ(t,d)

(U(t,Φ(t, d))

),U(t,Φ(t, d))〉V − 〈`(t),U(t,Φ(t, d))〉V

+α

2‖∇Φ(t, d)‖22 +

β

2‖Φ(t, d)− d‖22

= −1

2〈`(t),U(t,Φ(t, d))〉V +

α

2‖∇Φ(t, d)‖22 +

β

2‖Φ(t, d)− d‖22. (3.1)

This reformulation of the reduced energy allows us to show the following

Lemma 3.2 (Fréchet differentiability of I). It holds I ∈ C1([0, T ] × L2(Ω)) and, atall (t, d) ∈ [0, T ]× L2(Ω), we have

∂tI(t, d) = −〈 ˙(t),U(t,Φ(t, d))〉V , ∂dI(t, d) = β(d− Φ(t, d)). (3.2)

Proof. First note that the mapping

f : [0, T ]× L2(Ω)→ R, f(t, d) := 〈`(t),U(t,Φ(t, d))〉V

can be seen as product of the functions ` and [0, T ]×L2(Ω) 3 (t, d) 7→ U(t,Φ(t, d)) ∈V . The latter one is continuously Fréchet differentiable, thanks to Lemmas 2.11 and2.15. Together with Assumption 2.5 the product rule yields f ∈ C1([0, T ] × L2(Ω)).Thus, thanks to Lemma 2.15, we deduce from (3.1) that I ∈ C1([0, T ]× L2(Ω)) and,for given (t, d) ∈ [0, T ]× L2(Ω) and (δt, δd) ∈ R× L2(Ω), it holds

I ′(t, d)(δt, δd) = −1

2〈 ˙(t)δt,U(t,Φ(t, d))〉V −

1

2〈`(t),U ′(t,Φ(t, d))(δt, δϕ)〉V

+ α(∇Φ(t, d),∇δϕ)2 + β(Φ(t, d)− d, δϕ− δd)2,(3.3)

where we abbreviate δϕ = Φ′(t, d)(δt, δd). To derive the formulas for the partialderivatives, first observe that (2.4a) tested with U(t, ϕ), Definitions 2.6 and 2.9, andthe symmetry of C imply

〈 ˙(t),U(t,Φ(t, d))〉V = 〈AΦ(t,d)∂tU(t,Φ(t, d)),U(t,Φ(t, d))〉V= 〈`(t), ∂tU(t,Φ(t, d))〉V .

(3.4)

If one tests (2.4b) with U(t,Φ(t, d)) ∈ V , one further obtains

− 1

2〈`(t), ∂ϕU(t,Φ(t, d))δϕ〉V + α(∇Φ(t, d),∇δϕ)2 + β(Φ(t, d)− d, δϕ− δd)2

= −1

2〈div

(g′(Φ(t, d))(δϕ)Cε(U(t,Φ(t, d))

),U(t,Φ(t, d))〉V

+ α(∇Φ(t, d),∇δϕ)2 + β(Φ(t, d)− d, δϕ− δd)2

= 〈F(t,Φ(t, d)

)+BΦ(t, d), δϕ〉H1(Ω) − β(d, δϕ)2 + β(d− Φ(t, d), δd)2

= β(d− Φ(t, d), δd)2 ∀ δd ∈ L2(Ω),

(3.5)

where div : L2(Ω;RN×Nsym ) → V ∗ denotes the distributional divergence. Note thatthe last two equalities follow from (2.6), (2.7), and the definition of Φ, respectively.Inserting (3.4) and (3.5) in (3.3) leads to (3.2).

8

As an immediate consequence of Lemma 3.2 and the chain rule, one obtains thefollowing

Corollary 3.3 (Total derivative of I(·, d(·))). Let d ∈ C1([0, T ], L2(Ω)) be given.Then the map [0, T ] 3 t 7→ I(t, d(t)) is continuously differentiable with

d

dtI(t, d(t)) = ∂tI(t, d(t)) + (∂dI(t, d(t)), d(t))2 ∀ t ∈ [0, T ].

With the help of the reduced energy I we will deduce the energy balance from theevolutionary equation in (2.13). To this end note first that, due to the second equationin (3.2), the evolutionary equation (2.13) or equivalently (2.14) can also be written as

0 ∈ ∂Rδ(d(t)) + ∂dI(t, d(t)) ∀ t ∈ [0, T ], d(0) = d0. (3.6)

Since Rδ is proper and convex, this is in turn equivalent to

Rδ(d(t))+R∗δ(−∂dI(t, d(t))

)= (−∂dI(t, d(t)), d(t))2 ∀ t ∈ [0, T ], d(0) = d0, (3.7)

which will be the starting point for proving the energy balance in Proposition 3.5below. To summarize, we obtained the following four alternative, but yet equivalentformulations:

• the subdifferential formulations in (2.13) and (3.6), respectively,• the nonsmooth operator differential equation in (2.14),• the Fenchel-Young equality in (3.7).

In all what follows, we refer to these equivalent formulations simply as penalized dam-age evolution. Note that, since (2.13) and (2.14), respectively, are uniquely solvableby Theorem 2.16, the same holds for (3.6) and (3.7).

Lemma 3.4. If d satisfies the penalized damage evolution, then, for every t ∈ [0, T ],there holds R∗δ

(− ∂dI(t, d(t))

)= δ

2‖d(t)‖22.Proof. Let d satisfy the penalized damage evolution. Then it follows from (3.6) that∂Rδ(d(t)) 6= ∅ so that d ≥ 0. Hence, inserting (1.2) and (3.2) in (3.7) leads to

R∗δ(−∂dI(t, d(t))

)= (−β(d(t)−ϕ(t)), d(t))2−r‖d(t)‖1−

δ

2‖d(t)‖22 ∀ t ∈ [0, T ], (3.8)

where we again abbreviated ϕ = Φ(·, d(·)). From the equivalent formulation (2.14)multiplied with d(t) and integrated over Ω, we deduce that δ‖d(t)‖22 = (−β(d(t) −ϕ(t)), d(t))2 − r‖d(t)‖1. Inserting this into (3.8) gives the assertion.

Proposition 3.5 (Energy-dissipation balance). The unique solution d ∈ C1([0, T ];L2(Ω))of the penalized damage evolution fulfills for all 0 ≤ s ≤ t ≤ T the energy identity∫ t

s

Rδ(d(τ))dτ +

∫ t

s

R∗δ(− ∂dI(τ, d(τ))

)dτ + I(t, d(t))

= I(s, d(s)) +

∫ t

s

∂tI(τ, d(τ))dτ.(3.9)

Proof. Corollary 3.3 combined with (3.7) yields at all τ ∈ [0, T ] the identity

Rδ(d(τ)) +R∗δ(− ∂dI(τ, d(τ))

)= ∂tI(τ, d(τ))− d

dtI(τ, d(τ)). (3.10)

9

Recall that d ≥ 0 as a result of (2.14). This implies in view of (1.2) and d ∈C([0, T ], L2(Ω)) that the map [0, T ] 3 τ 7→ Rδ(d(τ)) ∈ R is continuous. From Lemma3.4 and Corollary 3.3 we deduce the continuity w.r.t. time of all terms in (3.10) andtherefore, the integrability thereof. Integrating (3.10) w.r.t. time then yields (3.9).Remark 3.6. One can show that the reverse statement of Proposition 3.5 is also true.Therefore, the energy balance is just another equivalent formulation of the penalizeddamage evolution. To do so, one combines (3.9) with Corollary 3.3 and Young’sinequality and in this way obtains (3.7), which was one of the equivalent formulationsof the penalized damage evolution. However, for the upcoming analysis we only needthe implication stated in Proposition 3.5 so that we do not go into more details.

4. Limit Analysis. This section proves the viability of the penalty approach,in the sense that one can pass to the limit β →∞ and in this way obtain a single-fielddamage model as stated in Section 5. In Section 7 below, we will then see that thelimit system is equivalent to a classical viscous partial damage model.In the first part of this section we focus on finding bounds independent of β in suitablespaces for the local and nonlocal damage, respectively. Note that, for the displace-ment, such a bound is already given in (2.2). This allows us to find weakly convergentsubsequences. The limiting behaviour thereof, as β → ∞, is studied in the secondand third part of this section.

4.1. Uniform Boundedness. For the rest of this subsection, we fix β > 0arbitrary so that Assumption 2.7.2 is fulfilled and denote the solution of (2.15) by(u, ϕ, d). We aim to derive bounds independent of β for (u, ϕ, d). The starting pointfor doing so is the energy-dissipation balance in Proposition 3.5. For this purposewe require the following additional assumption, which is rather self-evident in manypractical applications:Assumption 4.1. From now on we assume that at the beginning of the process thebody is completely sound, i.e. d0 ≡ 0, and that there is no load acting upon the bodyat initial time, i.e. `(0) ≡ 0.As a first consequence of Assumption 4.1, we obtain in view of (2.15) that

u(0) = ϕ(0) = d(0) ≡ 0. (4.1)

Lemma 4.2 (Boundedness of the local damage). Let Assumption 4.1 hold. Thenthere exists a constant C > 0, independent of β, such that ‖d‖H1(0,T ;L2(Ω)) ≤ C.Proof. The result follows mainly from the energy identity in Proposition 3.5. In orderto see this, set s := 0 and t = T in (3.9) and use (1.2), Lemma 3.4, (3.1), and (3.2),as well as Assumption 4.1, and (4.1) to obtain

δ

∫ T

0

‖d(τ)‖22 dτ =1

2〈`(T ),u(T )〉V +

∫ T

0

〈− ˙(τ),u(τ)〉V dτ

−(r

∫ T

0

‖d(τ)‖1 dτ +α

2‖∇ϕ(T )‖22 +

β

2‖ϕ(T )− d(T )‖22

)≤∫ T

0

‖ ˙(τ)‖V ∗‖u(τ)‖V dτ +1

2‖`(T )‖V ∗‖u(T )‖V ≤ C

with C > 0 independent of β. The assertion then follows from d0 = 0 and Poincaré-Friedrich’s inequality.

10

Next let us turn to the uniform boundedness of ϕ. We will establish the existence ofa constant C independent of β such that ‖ϕ‖H1(0,T ;H1(Ω)) ≤ C. In view of (4.1) andPoincaré-Friedrich’s inequality, we only need to show that there is C > 0 independentof β such that

‖ϕ‖2L2(0,T ;H1(Ω)) =

∫ T

0

‖ϕ(τ)‖22 dτ +

∫ T

0

‖∇ϕ(τ)‖22 dτ ≤ C. (4.2)

The starting point therefor is the equation characterizing the time derivative of thenonlocal damage. In view of Lemmata 2.12 and 2.15 this equation is given by

Bϕ(t) + ∂tF (t, ϕ(t)) + ∂ϕF (t, ϕ(t))ϕ(t) = βd(t) in H1(Ω)∗. (4.3)

Testing (4.3) with ϕ(t), integrating over [0, T ], and using (2.6) lead to

∫ T

0

α‖∇ϕ(τ)‖22 dτ = β

=: I1︷ ︸︸ ︷∫ T

0

(d(τ)− ϕ(τ), ϕ(τ))2 dτ

−∫ T

0

〈∂tF (t, ϕ(t)) + ∂ϕF (t, ϕ(t))ϕ(t), ϕ(t)〉 dτ︸ ︷︷ ︸=: I2

.

(4.4)

Lemma 4.3. Under Assumption 4.1 it holds I1 ≤ 0.Proof. We follow the ideas of [16, Proposition 4.3]. From Theorem 2.16 we recallthat d and ϕ are Lipschitz continuous, and therefore d ∈ W 1,∞(0, T ;L2(Ω)). Hence,by [30, Theorem 3.1.40], the mapping f : [0, T ]→ L2(Ω) defined through

f(t) := δd(t) + β(d(t)− ϕ(t)) + r (4.5)

is almost everywhere differentiable. Let now t ∈ (0, T ) be arbitrary, but fixed andh > 0 sufficiently small such that t+h ∈ (0, T ). From (2.14) it follows that d(τ, x) ≥ 0,f(τ, x) ≥ 0, and f(τ, x) d(τ, x) = 0 for all τ ∈ [0, T ] and almost all x ∈ Ω. Thus, wearrive at (f(t± h)− f(t)

h, d(t)

)2≥ 0.

Passing to the limit h 0 and keeping in mind the fact that f is almost every-where differentiable implies (f(t), d(t))2 = 0 f.a.a. t ∈ (0, T ). Thanks to (4.5) this isequivalent to δ(d(t), d(t))2 + (β(d(t)− ϕ(t)), d(t))2 = 0, which can be continued as

δ

2

d

dt‖d(t)‖22 + β‖d(t)− ϕ(t))‖22 + β(d(t)− ϕ(t), ϕ(t))2 = 0 (4.6)

for almost all t ∈ (0, T ). Due to Theorem 2.16, ϕ and d are both continuous withvalues in L2(Ω) so that we can integrate (4.6) over [0, T ]. This finally yields

δ

2‖d(T )‖22 −

δ

2‖d(0)‖22 + β

∫ T

0

‖d(τ)− ϕ(τ))‖22 dτ + β

∫ T

0

(d(τ)− ϕ(τ), ϕ(τ))2 dτ = 0,

which on account of (4.1) gives the assertion.Lemma 4.4. For all k > 0 it holds

|I2| ≤ c(k)

∫ T

0

‖ϕ(τ)‖2H1(Ω) dτ + k

∫ T

0

‖ϕ(τ)‖22 dτ + C k,

11

where c : R+ → R+ is a monotonically decreasing function, independent of β, whichtends to 0 as k →∞ and C > 0 is a constant independent of β.Proof. Let t ∈ [0, T ] be arbitrary, but fixed. From (2.9) we deduce

〈∂tF (t, ϕ(t)), ϕ(t)〉 = −〈div(g′(ϕ(t))ϕ(t)Cε(u(t))

), ∂tU(t, ϕ(t))〉V .

Due to p > N we have H1(Ω) → L2pp−2 (Ω) and thus, Hölder’s inequality with (p −

2)/2p+ 1/p+ 1/2 = 1 in combination with (2.2) and (2.5) yields∣∣〈∂tF (t, ϕ(t)), ϕ(t)〉H1(Ω)

∣∣ ≤ ‖g′(ϕ(t))‖∞‖ϕ(t)‖ 2pp−2‖u(t)‖W 1,p

D (Ω)‖∂tU(t, ϕ(t))‖V

≤ C‖ϕ(t)‖H1(Ω),

with C > 0 independent of β. On account of the generalized Young inequality, thiscan be continued as follows∣∣〈∂tF (t, ϕ(t)), ϕ(t)〉H1(Ω)

∣∣ ≤ 1

4k‖ϕ(t)‖2H1(Ω) + Ck ∀ k > 0. (4.7)

Together with (2.10) and the definition of I2 in (4.4), this gives the assertion withc(k) = 1

4k + c(k) so that c is indeed independent of β, monotonically decreasing andtends to 0 as k →∞.Lemma 4.5. Let Assumption 4.1 hold. Then there exist constants C1, C2 > 0, inde-pendent of β, such that∫ T

0

‖∇ϕ(τ)‖22 dτ ≤ C1

∫ T

0

‖ϕ(τ)‖22 dτ + C2.

Proof. Applying Lemmata 4.3 and 4.4 to the right-hand side in (4.4) yields

(α− c(k))

∫ T

0

‖∇ϕ(τ)‖22 dτ ≤ (c(k) + k)

∫ T

0

‖ϕ(τ)‖22 dτ + C k

for all k > 0. Since c tends to zero as k → ∞, there is a K > 0 such that c(K) < αholds. Choosing k = K thus yields the assertion. Note that K does not depend onβ, since c is independent of β.Lemma 4.6. Let Assumption 4.1 hold. Then, for β > 0 sufficiently large, there holds‖ϕ‖L2(0,T ;L2(Ω)) ≤ C with a constant C > 0 independent of β.Proof. From (4.4) we deduce∫ T

0

‖ϕ(τ)‖22 dτ ≤∫ T

0

(d(τ), ϕ(τ))2 dτ −1

βI2. (4.8)

Young inequality implies for the first term on the right-hand side in (4.8) that∫ T

0

(d(τ), ϕ(τ))2 dτ ≤1

2

∫ T

0

‖d(τ)‖22 dτ +1

2

∫ T

0

‖ϕ(τ)‖22 dτ. (4.9)

In order to estimate the second term on the right-hand side in (4.8), we apply Lemma4.4 for some fixed k > 0, which thanks to Lemma 4.5 gives

1

β|I2| ≤

C1

β

∫ T

0

‖ϕ(τ)‖22 dτ +C2

β(4.10)

12

with constants C1 and C2 independent of β on account of Lemmata 4.4 and 4.5.Inserting (4.9) and (4.10) in (4.8) then implies∫ T

0

‖ϕ(τ)‖22 dτ ≤1

2

∫ T

0

‖d(τ)‖22 dτ +(1

2+C1

β

)∫ T

0

‖ϕ(τ)‖22 dτ +C2

β. (4.11)

Now, for β sufficiently large such that C1/β < 1/2, the assertion follows from Lemma4.2.

As a consequence of Lemmata 4.5 and 4.6 and Poincaré-Friedrich’s inequality togetherwith (4.1) we can now state the main result of this section:

Corollary 4.7 (Boundedness of the nonlocal damage). Under Assumption 4.1 thereexists a β0 > 0 and a constant C > 0 such that ‖ϕ‖H1(0,T ;H1(Ω)) ≤ C for all β ≥ β0.

4.2. Passing to the Limit in the Elliptic System. We start our limit analysiswith the elliptic system (2.15a)–(2.15b). In order to emphasize the dependency on thepenalty parameter, we no longer suppress the index β and denote the unique solutionof (P) and (2.15a)–(2.15c), respectively, by (uβ , ϕβ , dβ).

Proposition 4.8 (Passing to the limit in (2.15a)). Let Assumption 4.1 hold. Then,for every sequence βn →∞, there exist a (not relabeled) subsequence ϕβnn∈N suchthat

ϕβn ϕ in H1(0, T ;H1(Ω)), (4.12)uβn = U(·, ϕβn(·))→ U(·, ϕ(·)) =: u in C([0, T ];V ) (4.13)

as n→∞.

Proof. Since H1(0, T ;H1(Ω)) is a reflexive Banach space, Corollary 4.7 implies theexistence of a (not relabeled) subsequence of ϕβnn∈N such that (4.12) holds.

To prove the second assertion, we apply Lemma 2.10 with π = 2. Then r = 2p/(p−2) and (2.3) implies that the mapping Uc : C([0, T ];Lr(Ω)) 3 ϕ 7→ U(·, ϕ(·)) ∈C([0, T ];V ) is Lipschitz continuous with constant L > 0. Note that L is independentof β, since β does not appear in the elliptic equation (2.15a) associated with U . Now,since p > 2 by Assumption 2.7.1, the embedding H1(Ω) →→ Lr(Ω) is compact,which implies that H1(0, T ;H1(Ω)) →→ C([0, T ];Lr(Ω)) is a compact as well, cf. [30,Corollary 3.1.42]. Consequently (4.12) leads to ϕβn → ϕ in C([0, T ];Lr(Ω)) and theLipschitz continuity of Uc then gives (4.13).

For the rest of this section we denote by βnn∈N a fixed sequence such that ϕβnconverges weakly in H1(0, T ;H1(Ω)) and by ϕ the limit of this particular sequence.Proposition 4.8 guarantees the existence of such a sequence. Notice however that(at this point) ϕ depends on the chosen subsequence. Nevertheless, as we will see inProposition 5.6 below, under a (rather restrictive) regularity condition on the ellipticoperator Aϕ, the weak limit is unique so that the whole sequence converges weakly.

Proposition 4.9 (Passing to the limit in (2.15b)). Let Assumption 4.1 hold andlet βnn∈N be the subsequence from Proposition 4.8 and ϕ the corresponding limit.Then there holds

dβn ϕ in H1(0, T ;L2(Ω)) as n→∞,

which implies in particular that both damage variables coincide in the limit.13

Proof. First of all, Lemma 4.2 yields the existence of a subsequence βnkk∈N so that

dβnk d in H1(0, T ;L2(Ω)) as k →∞. (4.14)

Now, let t ∈ [0, T ] and ψ ∈ H1(Ω) be arbitrary, but fixed. Testing (2.15b) with ψgives the following estimate, where we use Corollary 4.7, the boundedness of g′ fromAssumption 2.3, and (2.2):∫

Ω

(dβn(t)− ϕβn(t)

)ψ dx

≤ 1

βn

(α‖∇ϕβn(t)‖2 + ‖g′(ϕβn(t))‖∞‖Cε(uβn(t)) : ε(uβn(t))‖ p

2

)‖ψ‖H1(Ω)

≤ C

βn‖ψ‖H1(Ω) → 0 as n→∞.

(4.15)

Note that H1(0, T ;H1(Ω)) → C([0, T ];H1(Ω)), which yields the boundedness of‖∇ϕβn(t)‖2 uniformly in t. Of course the above estimate also holds for the sub-sequence βnkk∈N and hence, (4.14) and the convergence of ϕβn by assumptionimply ∫

Ω

(d(t)− ϕ(t))ψ dx ≤ 0,

and, since t and ψ were arbitrary, this gives in turn d(t) = ϕ(t) for all t ∈ [0, T ].Hence, we obtain dβnk ϕ in H1(0, T ;L2(Ω)) as k → ∞. Thus, the weak limit isunique and a well known argument implies the convergence of the whole sequencedβn (with βnn from Proposition 4.8) to ϕ.

4.3. Passing to the Limit in the Energy Balance. We now turn our at-tention to the passage to the limit in (2.15c). However, as already indicated at thebeginning of Section 3, the term β(d− ϕ) involved in (2.15c) is not bounded in suit-able spaces that allow for a passage to the limit. Instead we will pass to the limit in(3.9), which will result in an energy inequality, that turns out to be equivalent to anevolutionary equation as shown in Section 5.We begin by introducing the energy and dissipation functionals that will arise afterpassing to the limit. The energy without penalty term reads as followsDefinition 4.10 (Energy functionals without penalty). We define the energy func-tional without penalty term by

E : [0, T ]× V ×H1(Ω)→ R,

E(t,u, ϕ) :=1

2

∫Ω

g(ϕ)Cε(u) : ε(u) dx− 〈`(t),u〉V +α

2‖∇ϕ‖22.

The reduced energy functional without penalty is given by

I : [0, T ]×H1(Ω)→ R, I(t, ϕ) := E(t,U(t, ϕ), ϕ).

Remark 4.11. Since the expressions in E involving the displacement are not affectedby the penalty term, it follows that, for a given pair (t, ϕ) ∈ [0, T ]×H1(Ω), u solves

minu∈VE(t,u, ϕ),

14

iff u = U(t, ϕ) with U as defined in Definition 2.9. As a consequence we obtainI(t, ϕ) = minu∈V E(t,u, ϕ).

In complete analogy to (3.1), the definitions of E and U allow us to rewrite the reducedenergy functional without penalty as

I(t, ϕ) = −1

2〈`(t),U(t, ϕ)〉V +

α

2‖∇ϕ‖22. (4.16)

Lemma 4.12 (Fréchet differentiability of I). It holds I ∈ C1([0, T ]×H1(Ω)) and itspartial derivatives read

∂tI(t, ϕ) = −〈 ˙(t),U(t, ϕ)〉V , ∂ϕI(t, ϕ) = −α4ϕ+ F (t, ϕ), (4.17)

where 4 : H1(Ω)→ H1(Ω)∗ denotes the distributional Laplace operator.

Proof. The proof is completely along the lines of the proof of Lemma 3.2 so that weshorten the exposition. By applying the product rule to (4.16), one obtains that I isindeed continuously Fréchet-differentiable with

I ′(t, ϕ)(δt, δϕ) = −1

2〈 ˙(t)δt,U(t, ϕ)〉V −

1

2〈`(t),U ′(t, ϕ)(δt, δϕ)〉V + α(∇ϕ,∇δϕ)2.

Similarly to (3.4) and (3.5), the first to addends can be rewritten by using (2.4a), thesymmetry of C, (2.4b), and the definition of F to obtain

1

2〈 ˙(t)δt,U(t, ϕ)〉V +

1

2〈`(t),U ′(t, ϕ)(δt, δϕ)〉V = 〈 ˙(t)δt,U(t, ϕ)〉V −〈F (t, ϕ), δϕ〉H1(Ω),

which completes the proof.

Next we introduce the viscous dissipation functional corresponding to the situationwithout penalty:

Definition 4.13 (Viscous dissipation functional without penalty). We define thefunctional Rδ by

Rδ : H1(Ω)→ [0,∞], Rδ(η) :=

r∫

Ωη dx+ δ

2‖η‖22 if η ≥ 0 a.e. in Ω,

∞ otherwise.

Note that Rδ coincides with Rδ from (1.2) apart from its domain which is now H1(Ω)instead of L2(Ω). This is in accordance with the spatial regularity of the damagevariable in the limit, as seen in Proposition 4.8.

In order to pass to the limit in (3.9) we consider a sequence βn →∞ such that

ϕβn ϕ in H1(0, T ;H1(Ω)), (4.18)

dβn ϕ in H1(0, T ;L2(Ω)), (4.19)uβn → U(·, ϕ(·)) in C([0, T ];V ). (4.20)

Recall that such a sequence exists according to Propositions 4.8 and 4.9.

Lemma 4.14. Under Assumption 4.1 it holds for all t ∈ [0, T ] that∫ t

0

Rδ(ϕ(τ)) dτ ≤ lim infn→∞

∫ t

0

Rδ(dβn(τ)) dτ.

15

Proof. Let t ∈ (0, T ) be arbitrary, but fixed. From (4.19) it follows that

dβn ϕ in L2(0, t;L2(Ω)) (4.21)

so that dβn ≥ 0 a.e. in Ω × (0, t), see (2.15c), implies ϕ ≥ 0 a.e. in Ω × (0, t) bythe weak closedness of the set of non-negative functions in L2(0, t;L2(Ω)). Thus, thedefinition of Rδ implies∫ t

0

Rδ(ϕ(τ)) dτ = r ‖ϕ‖L1(0,t;L1(Ω)) +δ

2‖ϕ‖2L2(0,t;L2(Ω))

and the same obviously holds for Rδ(dβn(τ)), cf. (1.2). The result then follows fromthe weak lower semicontinuity of (squared) norms.

Lemma 4.15. Let Assumption 4.1 hold. Then for all t ∈ [0, T ] we have

∂dI(t, dβn(t)) ∂ϕI(t, ϕ(t)) in H1(Ω)∗ as n→∞.

Proof. Let t ∈ [0, T ] be arbitrary, but fixed and set again r = 2p/(p − 2). Asexplained at the end of the proof of Proposition 4.8, Assumption 2.7.1 implies thecompact embedding H1(0, T ;H1(Ω)) →→ C([0, T ];Lr(Ω)) so that (4.18) results in

ϕβn(t)→ ϕ(t) in Lr(Ω) for n→∞, ∀ t ∈ [0, T ]. (4.22)

Furthermore, since H1(0, T ;H1(Ω)) → C([0, T ];H1(Ω)), (4.18) implies

∇ϕβn(t) ∇ϕ(t) in L2(Ω) for n→∞, ∀ t ∈ [0, T ]. (4.23)

From (3.2) and (2.15b) we moreover deduce

∂dI(t, dβn(t)) = βn(dβn(t)− ϕβn(t)) = −α4ϕβn(t) + F (t, ϕβn(t)).

Together with (4.17) and (2.8) this yields for every v ∈ H1(Ω) that

|〈∂dI(t, dβn(t))− ∂ϕI(t, ϕ(t)), v〉H1(Ω)|≤ α|(∇ϕβn(t)−∇ϕ(t),∇v)2|+ |〈F (t, ϕβn(t))− F (t, ϕ(t)), v〉|≤ α|(∇ϕβn(t)−∇ϕ(t),∇v)2|+ C ‖ϕβn(t)− ϕ(t)‖r‖v‖r.

The result then follows from (4.22), (4.23), and H1(Ω) → Lr(Ω) by Assumption 2.7.1.

Next we turn to the conjugate functional of Rδ. According to Definition 4.13, R∗δ isdefined on H1(Ω)∗, which is essential as ∂ϕI(τ, ϕ(τ)) is only an element of H1(Ω)∗,cf. Lemma 4.15.

Lemma 4.16. Under Assumption 4.1 it holds for all t ∈ [0, T ]∫ t

0

R∗δ(− ∂ϕI(τ, ϕ(τ))

)dτ ≤ lim inf

n→∞

∫ t

0

R∗δ(−∂dI(τ, dβn(τ))) dτ.

16

Proof. Again, let t ∈ [0, T ] by arbitrary, but fixed. By definition of the Fenchel-conjugate, it holds for any ξ ∈ L2(Ω) that

R∗δ(ξ) = supv∈H1(Ω)

((ξ, v)2 − Rδ(v)

)≤ supv∈L2(Ω)

((ξ, v)2 −Rδ(v)

)= R∗δ(ξ). (4.24)

Notice that we used in the above estimate that Rδ and Rδ are defined with differentdomains, see (1.2) and Definition 4.13. Further, R∗δ : H1(Ω)∗ → (−∞,∞] is convexand lower semicontinuous and thus weakly lower semicontinuous, which thanks toLemma 4.15 leads to

R∗δ(− ∂ϕI(τ, ϕ(τ))

)≤ lim inf

n→∞R∗δ(− ∂dI(τ, dβn(τ))

)∀ τ ∈ [0, t].

By setting ξ := −∂dI(τ, dβn(τ)) ∈ L2(Ω), see (3.2), in (4.24), the above estimate canbe continued as

R∗δ(− ∂ϕI(τ, ϕ(τ))

)≤ lim inf

n→∞R∗δ(− ∂dI(τ, dβn(τ))

)= lim inf

n→∞

δ

2‖dβn(τ)‖22 (4.25)

for all τ ∈ [0, t], where the last equality follows from Lemma 3.4. Applying Fatou’slemma to the right-hand side gives∫ t

0

lim infn→∞

R∗δ(− ∂dI(τ, dβn(τ))

)dτ ≤ lim inf

n→∞

∫ t

0

R∗δ(− ∂dI(τ, dβn(τ))

)dτ. (4.26)

Furthermore, arguing analogously to the derivation of (4.23), one sees that (4.19)implies dβn(τ) ϕ(τ) in L2(Ω) for every τ ∈ [0, T ]. Thus, (4.25) shows that R∗δ

(−

∂ϕI(τ, ϕ(τ)))is finite for every τ . In addition, due to (4.17) and H1(0, T ;H1(Ω)) →

C([0, T ];H1(Ω)), the map [0, t] 3 τ 7→ ∂ϕI(τ, ϕ(τ)) ∈ H1(Ω)∗ is continuous. SinceR∗δ is lower semicontinuous, it thus follows that the mapping

[0, t] 3 τ 7→ R∗δ(−∂ϕI(τ, ϕ(τ))) ∈ R

is lower semicontinuous as well and therefore, measurable. Now we can integrate(4.25) over (0, t), which combined with (4.26) finally gives the assertion.

Proposition 4.17 (Energy inequality without penalty). Let Assumption 4.1 hold.Then the limit function ϕ ∈ H1(0, T ;H1(Ω)) fulfills for all t ∈ [0, T ] the estimate∫ t

0

Rδ(ϕ(τ)) dτ +

∫ t

0

R∗δ(− ∂ϕI(τ, ϕ(τ))

)dτ + I(t, ϕ(t))

≤ I(0, ϕ(0)) +

∫ t

0

∂tI(τ, ϕ(τ)) dτ.(4.27)

Proof. Let t ∈ [0, T ] be arbitrary, but fixed. Setting s := 0 in (3.9) yields∫ t

0

Rδ(dβn(τ)) dτ +

∫ t

0

R∗δ(− ∂dI(τ, dβn(τ))

)dτ + I(t, dβn(t))

= I(0, dβn(0)) +

∫ t

0

∂tI(τ, dβn(τ))

)dτ ∀n ∈ N.

(4.28)

17

In view of Lemmas 4.14 and 4.16 we only need to discuss the last three terms in theabove equation. To this end, we combine (3.1) and (4.16) with (4.20), (4.23), and theweakly lower semicontinuity of ‖ · ‖22, which gives

I(t, ϕ(t)) = −1

2〈`(t),U(t, ϕ(t))〉V +

α

2‖∇ϕ(t)‖22

≤ lim infn→∞

(− 1

2〈`(t),uβn(t)〉V +

α

2‖∇ϕβn(t)‖22 +

βn2‖ϕβn(t)− dβn(t)‖22︸ ︷︷ ︸

≥0

)= lim inf

n→∞I(t, dβn(t)),

i.e., the desired convergence of the last term on the left-hand side of (4.28). It remainsto discuss the right-hand side in (4.28). Thanks to Assumption 4.1 and (4.1) the initialvalue just vanishes, i.e.,

I(0, dβn(0)) = 0 ∀n ∈ N, (4.29)

and, in light of (4.16), `(0) = 0 by Assumption 4.1, and the pointwise convergence in(4.23), which gives ∇ϕ(0) = 0, we obtain the same for the limit, i.e., I(0, ϕ(0)) = 0.Therefore, the formulas for the partial derivatives of I and I in (3.2) and (4.17)together with the regularity of ` and the convergence of the displacement in (4.20)finally imply

I(0, dβn(0)) +

∫ t

0

∂tI(τ, dβn(τ))) dτ

=

∫ t

0

〈− ˙(τ),uβn(τ)〉 dτ →∫ t

0

〈− ˙(τ),U(τ, ϕ(τ))〉 dτ

= I(0, ϕ(0)) +

∫ t

0

∂tI(τ, ϕ(τ)) dτ

which completes the proof.

In the next sections we use the energy inequality in (4.27) to show that the limit in(4.18)–(4.20) satisfies a system of equations which is equivalent to a classical viscouspartial damage model containing only one single damage variable. As secondary resultwe will also see that the inequality (4.27) is in fact equivalent to an energy identity,see Remark 5.2 below.

5. A Single-Field Gradient Damage Model. In this section we show thatevery solution of the energy inequality (4.27) satisfies an evolutionary equation andvice versa. The proof mainly follows the arguments of [17, Proposition 3.1].

Proposition 5.1 (Viscous differential inclusion without penalty). Every functionϕ ∈ H1(0, T ;H1(Ω)) which fulfills for all t ∈ [0, T ] the energy inequality (4.27) alsosatisfies the following evolutionary equation

0 ∈ ∂Rδ(ϕ(t)) + ∂ϕI(t, ϕ(t)) in H1(Ω)?, f.a.a. t ∈ (0, T ). (5.1)

The reverse assertion is true as well.

Proof. We start the proof with two auxiliary results needed for both implicationsstated in the proposition. To this end let ϕ ∈ H1(0, T ;H1(Ω)) first be arbitrary, but

18

fixed. Since Rδ is convex and proper, a classical result from convex analysis leads to

Rδ(ϕ(t)) + R∗δ(−∂ϕI(t, ϕ(t))) = −〈∂ϕI(t, ϕ(t)), ϕ(t)〉H1(Ω) (5.2)⇐⇒

−∂ϕI(t, ϕ(t)) ∈ ∂Rδ(ϕ(t)) (5.3)

f.a.a. t ∈ (0, T ). Further note that (4.17), combined with (2.7), (2.2), and the bound-edness assumption on g′, implies

‖∂ϕI(t, v)‖H1(Ω)∗ ≤ C ‖v‖H1(Ω) + c ∀ (t, v) ∈ [0, T ]×H1(Ω), (5.4)

where C, c > 0 are independent of (t, v). By using the density of C1([0, T ];H1(Ω)) inH1(0, T ;H1(Ω)), see e.g. [27, Lemma 7.2], and the continuous Fréchet differentiabilityof I by Lemma 4.12, as well as (5.4), one shows that the function [0, T ] 3 t 7→I(t, ϕ(t)) ∈ R belongs to H1(0, T ) with weak derivative

d

dtI(., ϕ(.)) = ∂tI(., ϕ(.)) + 〈∂ϕI(., ϕ(.)), ϕ(.)〉H1(Ω) ∈ L2(0, T ). (5.5)

Note that ϕ ∈ H1(0, T ;H1(Ω)) → C([0, T ];H1(Ω)) and (5.4) imply ∂ϕI(., ϕ(.)) ∈L∞(0, T ;H1(Ω)∗), which in turn renders the L2-regularity of d

dt I(., ϕ(.)).

Let us now assume that ϕ fulfills (4.27) for all t ∈ [0, T ]. Due to I(·, ϕ(·)) ∈ H1(0, T )the energy inequality implies by setting t = T that∫ T

0

Rδ(ϕ(τ)) dτ +

∫ T

0

R∗δ(−∂ϕI(τ, ϕ(τ))) dτ

≤ −∫ T

0

( ddtI(τ, ϕ(τ))− ∂tI(τ, ϕ(τ))

)dτ = −

∫ T

0

〈∂ϕI(τ, ϕ(τ)), ϕ(τ)〉H1(Ω) dτ,

where we used (5.5) for the last equality. Combining this with Young’s inequality,i.e.,

Rδ(ϕ(t)) + R∗δ(−∂ϕI(t, ϕ(t))) ≥ −〈∂ϕI(t, ϕ(t)), ϕ(t)〉H1(Ω) f.a.a. t ∈ (0, T ),

leads to (5.2) and consequently (5.3), which shows the first implication.The reverse assertion can be concluded by following the lines of the proof of Propo-sition 3.5. To see this, assume that ϕ ∈ H1(0, T ;H1(Ω)) satisfies (5.1). From theequivalence (5.3)⇐⇒ (5.2) and (5.5) we then obtain

Rδ(ϕ(t))+R∗δ(−∂ϕI(t, ϕ(t))) = − d

dtI(t, ϕ(t))+∂tI(t, ϕ(t)) f.a.a. t ∈ (0, T ). (5.6)

Note that any ϕ, which fulfills (5.1), automatically satisfies ϕ ≥ 0 in view of Definition4.13. The latter one then also ensures the L1-integrability of Rδ(ϕ(·)). For the right-hand side in (5.6) we have due to Lemma 4.12 and (5.5) that ∂tI(·, ϕ(·)) ∈ C[0, T ]

and ddt I(·, ϕ(·)) ∈ L2(0, T ), respectively. Thus, we are allowed to integrate (5.6) in

time, which implies∫ t

0

Rδ(ϕ(τ)) dτ +

∫ t

0

R∗δ(−∂ϕI(τ, ϕ(τ))) dτ

= I(0, ϕ(0))− I(t, ϕ(t)) +

∫ t

0

∂tI(τ, ϕ(τ)) dτ(5.7)

19

for all t ∈ [0, T ]. This completes the proof.Remark 5.2. An inspection of the proof of Proposition 5.1 shows that, in order toprove (5.1), it suffices that (4.27) holds only at t = T . Moreover, the proof showsthat (4.27) implies (5.1) which in turn gives (5.7). In this way we have shown that(4.27) is indeed an energy identity (in form of an energy balance). Furthermore,integrating (5.6) over an arbitrary interval [s, t] ⊂ [0, T ] (instead of [0, t]) leads toan energy identity, completely analogous to (3.9) so that the passage to the limitβ →∞ indeed preserves the structure of the energy-dissipation balance. We also referto [17, Proposition 3.1].

We summarize our results so far in the followingTheorem 5.3 (Single-field damage model). Let Assumption 4.1 hold and βnn∈Nbe a sequence with βn → ∞ as n → ∞. Then there is a subsequence (denoted by thesame symbol) and a function ϕ ∈ H1(0, T ;H1(Ω)) such that

ϕβn ϕ in H1(0, T ;H1(Ω)), dβn ϕ in H1(0, T ;L2(Ω)),

uβn → u := U(·, ϕ(·)) in C([0, T ];V ).(5.8)

Moreover, every limit (ϕ,u) ∈ H1(0, T ;H1(Ω)) × C([0, T ];V ) of such a sequencesatisfies f.a.a. t ∈ (0, T ) the following PDE system:

−div g(ϕ(t))Cε(u(t)) = `(t) in V ∗, (5.9a)

δ ϕ(t)− α4ϕ(t) +1

2g′(ϕ(t))C ε(u(t)) : ε(u(t)) + ∂R1(ϕ(t)) ∈ 0, ϕ(0) = 0, (5.9b)

with the (non-viscous) dissipation potential R1 defined by

R1 : H1(Ω)→ [0,∞], R1(η) :=

r∫

Ωη dx, if η ≥ 0 a.e. in Ω,

∞, otherwise.(5.10)

Proof. The existence of the subsequence has already been established in Propositions4.8 and 4.9. Furthermore, (5.9a) is just equivalent to u := U(·, ϕ(·)). It remains toverify (5.9b), which follows from (5.1). To see this, just apply (4.17) and the definitionof F to the left-hand side of (5.1) and use the sum rule for convex subdifferentials forthe right-hand side.The above theorem shows that (5.9) admits at least one solution. Of course, it wouldbe desirable to have the uniqueness of the solution, too, in particular, since thisguarantees the uniqueness of the limit in (5.8) and thus the (weak) convergence ofthe whole sequence. Unfortunately, for this purpose, we have to require the followingrather restrictive assumption. We underline that this assumption is only needed toshow uniqueness, while the rest of the analysis remains unaffected, if it is not fulfilled.Assumption 5.4. To ensure uniqueness of the solution of (5.9), we require that thereexists some p > 4 in the two-dimensional case and p ≥ 6 in the three-dimensional casesuch that the operator Aϕ : W 1,p

D (Ω)→W−1,pD (Ω) is continuously invertible for every

ϕ ∈ H1(Ω) and the norm of its inverse is bounded uniformly w.r.t. ϕ.

Remark 5.5. Assumption 5.4 is fulfilled, provided that no mixed boundary conditionsare present, the domain is smooth enough, and the difference between the boundednessand ellipticity constants of the stress strain relation is sufficiently small, cf. [21, Re-mark 3.11] and [11, 13]. Adapted to our situation this means that the values ε γC

20

and ‖C‖∞ have to be sufficiently close to each other, which is clearly rather restric-tive (beside the smoothness assumption on the domain), cf. also Remark 2.8. Theseassumptions on the data can be weakened, if one uses Hs(Ω) with s > N/2 insteadH1(Ω) as function space for the nonlocal damage in the penalized model (P). Werefer to [17, Sections 2.4 and 3.2] for details. Since the bilinear form associated withHs(Ω) is harder to realize in numerical practice, we do not follow this approach.

Proposition 5.6 (Uniqueness under additional assumptions). Under Assumptions5.4, system (5.9) admits a unique solution (ϕ,u) ∈ H1(0, T ;H1(Ω))× C([0, T ];V ).

Proof. Let (ϕi,ui) ∈ H1(0, T ;H1(Ω))×C([0, T ];V ), i = 1, 2 be two solutions of (5.9).First note that, from the definition of F in (2.7) and ui(·) = U(·, ϕi(·)) we have thatϕi satisfies

δϕi(t)− α4ϕi(t) + F (t, ϕi(t)) ∈ −∂R1(ϕi(t)), i = 1, 2, (5.11)

f.a.a. t ∈ (0, T ). Therefore, ∂R1(ϕi(t)) 6= ∅, which gives ϕ1, ϕ2 ≥ 0 f.a.a. t ∈ (0, T ).By testing (5.11) for i = 1 with ϕ2 − ϕ1 and vice versa and adding the arisinginequalities, we arrive at

δ‖ϕ1(t)− ϕ2(t)‖22 + α(∇ϕ1(t)−∇ϕ2(t),∇ϕ1(t)−∇ϕ2(t))2

≤ 〈F (t, ϕ2(t))− F (t, ϕ1(t)), ϕ1(t)− ϕ2(t)〉H1(Ω) f.a.a. t ∈ (0, T ).

Then, adding α(ϕ1(t) − ϕ2(t), ϕ1(t) − ϕ2(t))2 on both sides of this estimate andapplying (2.8) with r = 2p/(p− 4) (such that s = 2) lead to

δ‖ϕ1(t)− ϕ2(t)‖22 + α(ϕ1(t)− ϕ2(t), ϕ1(t)− ϕ2(t))H1(Ω)

≤ C‖ϕ1(t)− ϕ2(t)‖H1(Ω)‖ϕ1(t)− ϕ2(t)‖2

≤ C

4ε‖ϕ1(t)− ϕ2(t)‖2H1(Ω) + Cε ‖ϕ1(t)− ϕ2(t)‖22 ∀ ε > 0,

where the last estimate follows from the generalized Young inequality. Moreover,we used H1(Ω) → L2p/(p−4)(Ω) by Assumption 5.4. By choosing ε := δ/(2C) weconclude

α(ϕ1(t)− ϕ2(t), ϕ1(t)− ϕ2(t))H1(Ω) ≤C2

2δ‖ϕ1(t)− ϕ2(t)‖2H1(Ω) (5.12)

f.a.a. t ∈ (0, T ). Chain rule combined with the fact that ϕ1−ϕ2 is almost everywheredifferentiable (see [4]) yields∫ t

0

(ϕ1(τ)− ϕ2(τ), ϕ1(τ)− ϕ2(τ))H1(Ω) dτ

=1

2‖ϕ1(t)− ϕ2(t)‖2H1(Ω) −

1

2‖ϕ1(0)− ϕ2(0)‖2H1(Ω) ∀ t ∈ [0, T ].

Due to ϕ1(0) = ϕ2(0), we obtain after integrating (5.12) that

α

2‖ϕ1(t)− ϕ2(t)‖2H1(Ω) ≤

C2

2δ

∫ t

0

‖ϕ1(τ)− ϕ2(τ)‖2H1(Ω) dτ ∀ t ∈ [0, T ],

which by means of Gronwall’s lemma leads to

‖ϕ1(t)− ϕ2(t)‖2H1(Ω) ≤ 0 ∀ t ∈ [0, T ], (5.13)21

and thus, completes the proof.

As an immediate consequence of the uniqueness result we obtain the following

Corollary 5.7. If Assumption 5.4 is fulfilled, then the convergence in (5.8) is notonly valid for a subsequence, but for the whole sequence (dβn , ϕβn ,uβn).

6. Uniform Pointwise Bounds for the Damage Variable. This section isdevoted to deriving L∞-bounds for the damage variables dβ and ϕβ that are inde-pendent of the penalty parameter β. In view of (4.18) and (4.19), respectively, thisboundedness result carries over to the limit damage variable ϕ. These estimates arecrucial for the transformation of our damage model in (5.9) into another one that canbe seen as a classical viscous damage model, which will be performed in the upcomingSection 7. We emphasize that the Lipschitz continuity result in [21, Theorem 4.6]together with Sobolev embeddings already implies that the nonlocal damage ϕβ ispointwisely bounded in space and time, at least in two-dimensions, but these boundsmay depend on β. To show the desired boundedness independent of β, we mostlyfollow the ideas of [17], where the authors derive a similar result in two dimensionsvia time discretization, see [17, Proposition 4.2].

In the sequel, let β > 0 be fixed, but arbitrary so that Assumption 2.7 is fulfilled. Westart with the operator differential equation for the local damage variable in (2.14),which serves as the basis for the time discretization:

d(t) =1

δmax

− β

(d(t)− Φ(t, d(t))

)− r, 0

∀ t ∈ [0, T ], d(0) = d0, (6.1)

Here we suppressed again the dependency of d on β and simply write d instead ofdβ . To introduce a time-discrete incremental problem, let a number of time-stepsn ∈ N+ be given, set τ := T/n, and denote by tτk = kτk=0,...,n the correspondingpartition of the time interval [0, T ]. Further we define, beginning with dτ0 := d0, theapproximation of the local damage at time point tτk+1, k ∈ 0, ..., n−1, as the uniquesolution of the fixed-point equation

dτk+1 = dτk +τ

δmax

− β(dτk+1 − Φ(tτk+1, d

τk+1))− r, 0

. (Pβk)

Note that the unique solvability of (Pβk) is ensured for τ > 0 sufficiently small, as aresult of the Lipschitz continuity of max and Φ, cf. Lemma 2.14, see also [8, Thm.7.5.3]. In the following, we always tacitly assume that τ > 0 is chosen sufficientlysmall so that (Pβk) is uniquely solvable. We introduce the notations

tτ (t) := tτk+1 for t ∈ (tτk, tτk+1], k ∈ 0, ..., n− 1, tτ (0) := 0,

and we define the piecewise constant interpolating functions dτ : [0, T ]→ L2(Ω) by

dτ (t) := dτk+1 for t ∈ (tτk, tτk+1], k ∈ 0, ..., n− 1, dτ (0) := d0.

The piecewise linear interpolation dτ : [0, T ]→ L2(Ω) is given by

dτ (t) := dτk +t− tτkτ

(dτk+1 − dτk) for t ∈ [tτk, tτk+1], k ∈ 0, ..., n− 1.

22

Notice that dτ is differentiable on [0, T ] \ t0, t1, ..., tn with dτ (t) =dτk+1−d

τk

τ for t ∈(tτk, t

τk+1), k ∈ 0, ..., n− 1, so that (Pβk) implies for t ∈ [0, T ] \ t0, t1, ..., tn that

dτ (t) =1

δmax−β

(dτ (t)− Φ(tτ (t), dτ (t))

)− r, 0

. (6.2)

Proposition 6.1 (Convergence of the time-discretization). There holds

dτ → d in L∞(0, T ;L2(Ω)),

ϕτ := Φ(tτ (·), dτ (·))→ ϕ := Φ(·, d(·)

)in L∞(0, T ;H1(Ω)),

dτ → d in W 1,∞(0, T ;L2(Ω)),

as τ 0, where d ∈ C1,1([0, T ];L2(Ω)) is the solution to (6.1).

Proof. The first convergence is a result of [8, Thm. 7.5.3]. To see this, we recallthat the mapping [0, T ]× L2(Ω) 3 (t, η) 7→ 1

δ max− β

(η − Φ(t, η)

)− r, 0

∈ L2(Ω)

is Lipschitz continuous, by the Lipschitz continuity of max and Φ, cf. Lemma 2.14.Moreover, we note that d ∈ W 2,∞(0, T ;L2(Ω)). Thus, by [8, Thm. 7.5.3] and theLipschitz continuity of d, we have dτ → d in L∞(0, T ;L2(Ω)). By employing againthe Lipschitz continuity of Φ, we deduce

‖Φ(tτ (t), dτ (t)

)− Φ

(t, d(t)

)‖H1(Ω) ≤ K

(τ + ‖dτ − d‖L∞(0,T ;L2(Ω))

)∀ t ∈ [0, T ],

whence ϕτ → ϕ in L∞(0, T ;H1(Ω)) as τ 0 follows. This allows us to conclude thatdτ → d in W 1,∞(0, T ;L2(Ω)), in view of (6.1) and (6.2). The proof is now complete.

Next, we show that the piecewise constant interpolation function dτ is bounded a.e.in (0, T ) × Ω by a constant independent of β and τ . Because of Proposition 6.1 andTheorem 5.3, this bound carries over to the limit function ϕ in (5.9), which is theultimate goal of this section. With a little abuse of notation, we again suppress thesubscript β for the solution of (6.1), i.e., the problem with penalty, in order to shortenthe notation. When it comes to the pointwise boundedness of the “penalty limit” ϕ,i.e., the solution of (5.9), we will introduce the index β again, see Proposition 6.8below.

To prove the result, we follow the ideas of the proof of [17, Proposition 4.2], that is,we show the pointwise boundedness of dτk+1 by induction on the index k, see proofof Lemma 6.6 below. Therefor we first rewrite (Pβk) as an equivalent minimizationproblem:

Lemma 6.2. The solution dτk+1 of (Pβk) is the unique minimizer of

minv∈Ck

fk(v) := I(tτk+1, v) + r

∫Ω

v − dτk dx+δ

2

‖v − dτk‖22τ

(6.3)

with Ck := v ∈ L2(Ω) : v ≥ dτk a.e. in Ω.Proof. We aim to show that the necessary optimality conditions for (6.3) are equivalentto (Pβk). However, due to the nonlinear structure of U and Φ, the objective fk is ingeneral not convex so that it is a priori not clear that its necessary conditions arealso sufficient. It does therefore not suffice to prove the equivalence of (Pβk) to thenecessary conditions of (6.3) in order to establish the claim of the lemma. Instead

23

one first has to verify the existence of solutions for (6.3). To this end, first observethat, in view of (3.1),

fk(v) = −1

2〈`(tτk+1),U(tτk+1,Φ(tτk+1, v))〉V +

α

2‖∇Φ(tτk+1, v)‖22

+β

2‖Φ(tτk+1, v)− v‖22 + r

∫Ω

v − dτk dx+δ

2

‖v − dτk‖22τ

.(6.4)

The existence of solutions for (6.3) follows from the classical direct method of vari-ational calculus. First, Ck is convex and closed, thus, weakly closed. Moreover,in the light of (6.4) and (2.2), f is radially unbounded. It remains to show thatit is also weakly lower semicontinuous. To see this, we first prove that L2(Ω) 3v 7→ Φ(tτk+1, v) ∈ H1(Ω) is weakly continuous. To this end, consider a sequencevj ⊂ L2(Ω) such that vj v in L2(Ω) as j →∞ and abbreviate ϕj := Φ(tτk+1, vj)for the sake of convenience. The boundedness of vj and the global Lipschitz continu-ity of Φ by Lemma 2.14 imply the existence of a subsequence, for simplicity denoted bythe same symbol, such that ϕj ϕ in H1(Ω) as j →∞. Since p > N by Assumption2.7.1, H1(Ω) is compactly embedded in L2p/(p−2)(Ω) so that (2.8) implies

‖F (tτk+1, ϕj)− F (tτk+1, ϕ)‖H1(Ω)∗ ≤ C‖ϕj − ϕ‖2p/(p−2) → 0 as j →∞.

Together with the linearity of B, see (2.6), and the definition of Φ in Lemma 2.13,this yields

0 = β vj −Bϕj − F (tτk+1, ϕj) β v −Bϕ− F (tτk+1, ϕ) in H1(Ω)∗ as j →∞

so that ϕ = Φ(tτk+1, v). Thus, the weak limit is unique, which in turn gives theweak convergence of the whole sequence ϕj and, since vj was an arbitrary weaklyconverging sequence, we thus obtain the desired weak continuity of Φ(tτk+1, ·). To-gether with the weak lower semicontinuity of squared norms this gives the weak lowersemicontinuity of all terms on the right-hand side of (6.4), except the first addend.However, since the mapping L2p/(p−2)(Ω) 3 ϕ 7→ U(tτk+1, ϕ) ∈ V is continuous byLemma 2.10, the compactness of H1(Ω) → L2p/(p−2)(Ω) guarantees the convergenceof this term, too. Therefore, fk : L2(Ω) → R is weakly lower semicontinuous andconsequently, (6.3) admits solutions.Next we derive first-order necessary optimality conditions for (6.3). To this end, letz be an arbitrary, but fixed, solution thereof. Thanks to Lemma 3.2, fk is Fréchet-differentiable so that, due to convexity of Ck, the necessary optimality conditions for(6.3) are given by

f ′k(z)(v − z) ≥ 0 ∀ v ∈ Ck,

which, in view of (3.2), reads(β(z − Φ(tτk+1, z)) + r + δ τ−1(z − dτk) , v − δ τ−1(z − dτk)

)2≥ 0

for all v ∈ L2(Ω) with v ≥ 0 a.e. in Ω. (6.5)

As a variational inequality in L2(Ω) subject to pointwise inequality constraints, (6.5)can equivalently be reformulated as a pointwise complementarity relation (cf. e.g. [29,Section 2.8.2]), which reads

0 ≤ δ

τ(z − dτk) ⊥ β(z − Φ(tτk+1, z)) + r +

δ

τ(z − dτk) ≥ 0 a.e. in Ω.

24

Since the max-function is a well-known complementarity function, we infer that zsolves (Pβk) and, since the latter is uniquely solvable, this gives the uniqueness of theminimizer of (6.3) as well as the equivalence of (6.3) and (Pβk).

In order to obtain the desired boundedness result, we impose the following

Assumption 6.3. There exists a constant M > 0 such that g(x) ≥ g(M) for allx ≥M .

Remark 6.4. Recall that the function g : R → [ε, 1] measures the material rigidityof the body, where, according to Assumption 2.3, ε > 0 (which turns the model intoa partial damage model). Thus, there is always a minimum rigidity remaining, andAssumption 6.3 is for instance fulfilled, if g(x) ≡ ε for all x ≥ M , which means thatthis minimum rigidity is already achieved with finite values of the (nonlocal) damagevariable.

Next, let us introduce the Nemytskii operator associated with min·,M : R → R,denoted by L2(Ω) 3 z 7→ min(z,M) ∈ L2(Ω). Clearly, this operator is Lipschitzcontinuous with constant one. We will also consider this operator with domain andrange in H1(Ω), for simplicity denoted by the same symbol. It is well known thatmin(·,M) maps H1(Ω) to H1(Ω) with

∇min(v,M) = χv<M∇v ∀ v ∈ H1(Ω), (6.6)

where χv<m ∈ L∞(Ω; 0, 1) denotes the characteristic function of x ∈ Ω : v(x) <M (defined up to sets of measure zero), see e.g. [14, Theorem II.A.1].

Lemma 6.5. Under Assumption 6.3, we have

I(t,min(z,M)) ≤ I(t, z) ∀ (t, z) ∈ [0, T ]× L2(Ω).

Moreover, if z ∈ L2(Ω) satisfies min(z,M) = z, then min(Φ(t, z),M) = Φ(t, z) forevery t ∈ [0, T ].

Proof. Let (t, z) ∈ [0, T ] × L2(Ω) be arbitrary, but fixed and let us for simplicityabbreviate ϕ := Φ(t, z) in what follows. From Lemma 2.13, Definition 3.1, and (1.1),it follows that

I(t,min(z,M))

≤ E(t,U(t, ϕ),min(ϕ,M),min(z,M))

=1

2

∫Ω

g(min(ϕ,M))Cε(U(t, ϕ)) : ε(U(t, ϕ)) dx− 〈`(t),U(t, ϕ)〉V

+α

2‖∇min(ϕ,M)‖22 +

β

2‖min(ϕ,M)−min(z,M)‖22,

(6.7)

By Assumption 6.3, we further have g(min(ϕ,M)) ≤ g(ϕ) a.e. in Ω. Together with(2.1), (6.6), and the Lipschitz continuity of min(·,M) : L2(Ω) → L2(Ω), this impliesfor (6.7)

I(t,min(z,M)) ≤ 1

2

∫Ω

g(ϕ)Cε(U(t, ϕ)) : ε(U(t, ϕ)) dx

− 〈`(t),U(t, ϕ)〉V +α

2‖∇ϕ‖22 +

β

2‖ϕ− z‖22 = I(t, z),

(6.8)

where we again employed Definition 3.1. This proves the first assertion.25

Now, if z satisfies min(z,M) = z, then (6.8) holds with equality and thus, the in-equality in (6.7) is an equality, too. However, since the minimization problem in(P) is uniquely solvable, cf. Lemma 2.13, equality in (6.7) implies min(ϕ,M) =Φ(t,min(z,M)) = Φ(t, z), which completes the proof.

Lemma 6.6 (Uniform estimates for the discrete local and nonlocal damage). Supposethat Assumption 6.3 hold true and that d0 ∈ L∞(Ω) with ‖d0‖L∞(Ω) ≤ M . Then forall t ∈ [0, T ] there holds

dτ (t, x) ≤M, ϕτ (t, x) ≤M a.e. in Ω,

where ϕτ again stands for Φ(tτ (·), dτ (·)).

Proof. We follow the lines of the proof of [17, Proposition 4.2] and show that dτk ≤Mfor all k ∈ 0, ..., n by induction on the index k. Note that the assertion is fulfilledfor k = 0, since d0(x) ≤ M a.e. in Ω by assumption. Now let k ∈ 0, ..., n − 1 befixed, but arbitrary and assume that

dτk(x) ≤M a.e. in Ω. (6.9)

The idea of the proof is to show that (dτk+1)− := min(dτk+1,M) solves the problem(6.3) so that Lemma 6.2 implies dτk+1 = min(dτk+1,M). From (Pβk) it is clear thatdτk+1 ≥ dτk and, as a result of (6.9), thus

(dτk+1)− ≥ dτk a.e. in Ω ⇐⇒ (dτk+1)− ∈ Ck

so that (dτk+1)− is feasible for (6.3). For the objective of (6.3) we obtain by Lemma6.5 that

fk((dτk+1)−) = I(tτk+1, (dτk+1)−) + r

∫Ω

(dτk+1)− − dτk dx+δ

2

‖(dτk+1)− − dτk‖22τ

≤ I(tτk+1, dτk+1) + r

∫Ω

dτk+1 − dτk dx+δ

2

‖dτk+1 − dτk‖22τ

= fk(dτk+1)

where we also used that 0 ≤ (dτk+1)− − dτk ≤ dτk+1 − dτk. On the other hand, Lemma6.2 tells us that dτk+1 is the unique solution of (6.3) and, since (dτk+1)− is feasible asseen above, we conclude that (dτk+1)− = dτk+1. Hence, dτk+1 ≤ M , which ends theinduction step. Therefore, the piecewise constant interpolant satisfies dτ (t, x) ≤ Mfor all t ∈ [0, T ] and almost all x ∈ Ω. This, together with the second assertion ofLemma 6.5, implies ϕτ (t, x) = Φ(tτ (t), dτ (t)

)(x) ≤M for all t ∈ [0, T ] and almost all

x ∈ Ω, which completes the proof.

Remark 6.7. Let us point out that we are able to prove L∞-bounds for the discretizeddamage variables in both dimensions, while in [17, Proposition 4.2] the assertion isrestricted to the two-dimensional case. This is due to the fact that we consider theH1-seminorm of the (nonlocal) damage in the energy functional (1.1) if N ∈ 2, 3.In [17], this is the case only in two dimensions, while in three dimensions the authorswork with the H3/2-seminorm, see [17, Section 2.1]. In the latter case, there is noanalogue to the Stampacchia-lemma for H1-functions, which was essential for theproof of Lemma 6.5, see also [17, Eq. (4.30)]. We underline however that we areonly able to work with the H1-seminorm in the energy in both dimensions, because weimpose Assumption 2.7.1, see also Remark 2.8.

26

From the above results it easily follows that the local and the nonlocal damage associ-ated with the penalized problem (2.15), as well as their limit as β →∞, are boundeda.e. in space and time by the constant M from Assumption 6.3. To distinguish be-tween the solutions of the penalized problem and the single-field model in (5.9), weno longer suppress the index β.Proposition 6.8 (Uniform L∞-bound for the penalized damage variable). Let As-sumptions 4.1 and 6.3 be fulfilled. Then, the local and nonlocal damage associatedwith the penalized model (2.15) fulfill 0 ≤ dβ(t, x) ≤ M and ϕβ(t, x) ≤ M a.e. in(0, T )× Ω.

Proof. The lower bound for dβ is a direct consequence of (2.15c) and Assumption 4.1,i.e., d0 ≡ 0. The upper bounds immediately follow from Lemma 6.6, Proposition 6.1,and the closedness of v ∈ L2((0, T )× Ω) : v ≤M a.e. in (0, T )× Ω).By analogous arguments, we finally obtain the followingTheorem 6.9 (L∞-bound for the limit damage variable). Suppose that Assumptions4.1 and 6.3 are fulfilled. Let βn →∞ be a sequence so that dβn ϕ in H1(0, T ;L2(Ω))(whose existence is guaranteed by Theorem 5.3). Then there holds 0 ≤ ϕ ≤M a.e. in(0, T )× Ω, where M > 0 is the constant from Assumption 6.3.

7. Comparison to a Classical Partial Damage Model. The aim of thissection is to transfer the single-field damage model (5.9) (which describes the damageevolution in terms of a variable ϕ ∈ [0,∞)) to a system in terms of a damage variablez ∈ [0, 1]. This will allow for a comparison with the damage model in [17], which fallsinto the category of classical partial damage models, see e.g. [10]. Throughout thissection, we suppose that, beside our standing assumptions in Section 2, Assumptions4.1 and 6.3 are fulfilled, too. Let us pick a fixed, but arbitrary solution (ϕ,u) ∈H1(0, T ;H1(Ω)) × C([0, T ];V ) of (5.9), which arises as a penalty limit for β → ∞.Recall again that the existence of such a solution is guaranteed by Theorem 5.3.Moreover, we know from Theorem 6.9 that ϕ ∈ L∞((0, T )× Ω) with

0 ≤ ϕ(t, x) ≤M a.e. in (0, T )× Ω. (7.1)

The transformation of the single-field model (5.9) is based on the introduction of anew damage variable, defined by

z := 1− ϕ

M∈ H1(0, T ;H1(Ω)). (7.2)

Because of (7.1) and (7.2), it holds

z(t, x) ∈ [0, 1], z(t, x) = 1 ⇐⇒ ϕ(t, x) = 0, z(t, x) = 0 ⇐⇒ ϕ(t, x) = M

a.e. in (0, T )×Ω. Physically interpreted, this means that the body is completely sound,if z = 1, whereas the maximum damage is reached, if z = 0, cf. also Remark 6.4.We now transform the system (5.9) into a set of equations in the variables (z,u),i.e., the displacement remains the same. For this purpose, we define the followingtransformation of the elasticity coefficient function

g : R 3 x 7→ g(M(1− x)) ∈ R, (7.3)

as well as the following modified model parameters

α := αM2, δ := δM2, κ := rM. (7.4)27

Given these definitions, (5.9) is equivalent to

−div g(z(t))Cε(u(t)) = `(t) in V ∗, (7.5a)

δ z(t)− α4z(t) +1

2g′(z(t))C ε(u(t)) : ε(u(t)) ∈ −∂R1(z(t)), z(0) = 1 (7.5b)

with the dissipation functional

R1 : H1(Ω)→ [0,∞], R1(η) :=

κ∫

Ω(−η) dx, if η ≤ 0 a.e. in Ω,

∞, otherwise.(7.6)

Here we used that ∂R1(z(t)) = −M∂R1(ϕ(t)) f.a.a. t ∈ (0, T ). The energy functionalassociated with (7.5) reads

E(t,u, z) :=1

2

∫Ω

g(z)Cε(u) : ε(u) dx− 〈`(t),u〉V +α

2‖∇z‖22. (7.7)

Remark 7.1. The energy in (7.7) and the dissipation functional in (7.6) in principlecoincide with those in [17, Eq. (1.1) and (1.3)]. The major differences between E andthe energy in [17, Eq. (1.1)] concern the H1

0 -semi-norm in (7.7) and an additionalnonlinearity in [17, Eq. (1.1)]. This nonlinearity is just a smooth Nemytskii-operatoracting on z and can easily be incorporated into our analysis. We however decidednot to consider this nonlinearity and to rely on the energy from [5] instead, as thisreference serves as a basis for our two-field damage model.

In the two-dimensional case, the energy in [17] also contains the H10 -semi-norm, while,

in three spatial dimensions, the H10 -semi-norm is replaced by a bilinear form inducing

the H3/2-semi-norm in order to avoid Assumption 2.7.1, cf. Remark 2.8 and [17,Section 2.1].

Similarly to Lemma 2.13 one shows that, for given (t, z) ∈ [0, T ] × H1(Ω), the en-ergy E(t, ·, z) is minimized by the unique solution of (7.5a), see also [17, Lemma 2.1].Analogously to Definitions 2.9 and 3.1 we introduce the solution operator U : [0, T ]×H1(Ω) 3 (t, z) 7→ u ∈W 1,p(Ω) associated with (7.5a) and the reduced energy func-tional I(t, z) := E(t,U(t, z), z). Following the lines of the Lemmata 3.2 and 4.12,one shows that I is Fréchet-differentiable, cf. also [17, Corollary 2.1]. If we moreoverintroduce a viscous dissipation functional analogously to Definition 4.13 by

Rδ : H1(Ω) 3 η 7→ R1 +δ

2‖η‖22 ∈ R ∪ ∞,

then (7.5) is equivalent to

∂Rδ(z(t)) + ∂zI(t, z(t)) 3 0 in H1(Ω)?, f.a.a. t ∈ (0, T ). (7.8)

This is exactly the viscous model of [17], see [17, Eq. (3.1)], except for the differencesin the energy functional mentioned in Remark 7.1. This shows that the limit system(5.9) indeed coincides with the “classical” viscous partial damage model from [17],which may be seen as an ultimate argument showing that the penalization proceduremakes sense from a mathematical point of view.

A crucial result in [17] concerns the vanishing viscosity analysis for δ 0. It turnsout that the vanishing viscosity limit satisfies the following re-parametrized system

28

w.r.t. an artificial time s ∈ [0, S]

∂R1(z′(s)) + λ(s) z′(s) + ∂zI(t(s), z(s)) 3 0 f.a.a. s ∈ (0, S)

t(S) = T, t′(s) ≥ 0, t′(s)λ(s) = 0, t′(s) + |z′(s)| ≤ 1 a.e. in (0, S).

For details, we refer to [17, Section 5]. It is open question what happens to thepenalized model (P) and (2.15), respectively, when the viscosity vanishes, i.e., δ 0for fixed β > 0. This is of particular interest, since the non-viscous two-field model isfrequently considered in computational mechanics, see [5, 6]. The vanishing viscositylimit analysis for (P) however would go beyond the scope of this paper and gives riseto future research.

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